q-learning.tex

\documentclass{article}
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\usepackage{caption}
\usepackage{algorithm}
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\DeclareMathOperator*{\argmax}{arg\,max}

\begin{document}
\begin{preview}
    \begin{algorithm}[H]
        \begin{algorithmic}
        \Require
        \Statex Sates $\mathcal{X} = \{1, \dots, n_x\}$
        \Statex Actions $\mathcal{A} = \{1, \dots, n_a\},\qquad A: \mathcal{X} \Rightarrow \mathcal{A}$
        \Statex Reward function $R: \mathcal{X} \times \mathcal{A} \rightarrow \mathbb{R}$
        \Statex Black-box (probabilistic) transition function $T: \mathcal{X} \times \mathcal{A} \rightarrow \mathcal{X}$
        \Statex Learning rate $\alpha \in [0, 1]$, typically $\alpha = 0.1$
        \Statex Discounting factor $\gamma \in [0, 1]$
        \Procedure{QLearning}{$\mathcal{X}$, $A$, $R$, $T$, $\alpha$, $\gamma$}
            \State Initialize $Q: \mathcal{X} \times \mathcal{A} \rightarrow \mathbb{R}$ arbitrarily
            \While{$Q$ is not converged}
                \State Start in state $s \in \mathcal{X}$
                \While{$s$ is not terminal}
                    \State Calculate $\pi$ according to Q and exploration strategy (e.g. $\pi(x) \gets \argmax_{a} Q(x, a)$)
                    \State $a \gets \pi(s)$
                    \State $r \gets R(s, a)$ \Comment{Receive the reward}
                    \State $s' \gets T(s, a)$ \Comment{Receive the new state}
                    \State $Q(s', a) \gets (1 - \alpha) \cdot Q(s, a) + \alpha \cdot (r + \gamma \cdot \max_{a'} Q(s', a'))$
                    \State $s \gets s'$
                \EndWhile
            \EndWhile
            \Return $Q$
        \EndProcedure
        \end{algorithmic}
    \caption{$Q$-learning: Learn function $Q: \mathcal{X} \times \mathcal{A} \rightarrow \mathbb{R}$}
    \label{alg:q-learning}
    \end{algorithm}
\end{preview}
\end{document}